રેખીય સમય-અચલ (LTI) પ્રણાલીઓમાં કન્વોલ્યુશનના ગાણિતિક ગુણધર્મો અંગે નીચેનાં વિધાનો ધ્યાનમાં લો:
1. બે સંકેતો વચ્ચેની કન્વોલ્યુશન ક્રિયા સંકેતના પ્રકારને ધ્યાનમાં લીધા વિના ક્રમનો નિયમ (commutative) સંતોષે છે.
2. કન્વોલ્યુશનનો જૂથનો નિયમ (associative) અંતિમ આઉટપુટને અસર કર્યા વિના સંકેતોને જૂથમાં મૂકવા દે છે.
3. કન્વોલ્યુશન સરવાળા પર વિભાજિત થાય છે, જેથી જટિલ ઇનપુટને સરળ ઘટકોમાં વિભાજિત કરી શકાય.
4. કન્વોલ્યુશન ફક્ત સમય-અચલ પ્રણાલીઓને લાગુ પડે છે, સમય-ચલ પ્રણાલીઓને નહીં.
ઉપરોક્તમાંથી કયું/કયાં વિધાન સાચું/સાચાં છે?
જવાબ અને પગલાવાર ઉકેલ
B — ફક્ત 1, 2 અને 3. કન્વોલ્યુશન ક્રમનો નિયમ (x*h = h*x), જૂથનો નિયમ ((x*h₁)*h₂ = x*(h₁*h₂)) અને વિભાજનનો નિયમ (x*(h₁+h₂) = x*h₁ + x*h₂) સંતોષે છે. કન્વોલ્યુશન એક ગાણિતિક ક્રિયા છે જે સમય-ચલ સંદર્ભોમાં પણ વ્યાખ્યાયિત કરી શકાય છે, તેથી વિધાન 4 ખોટું છે.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
The convolution operation between two signals satisfies the commutative property irrespective of signal type.
Convolution is commutative, associative and distributive. - Statement 2 - Correct.
The associative property of convolution allows grouping of signals without affecting the final output.
Convolution is commutative, associative and distributive. - Statement 3 - Correct.
Convolution distributes over addition, enabling decomposition of complex inputs into simpler components.
Convolution is commutative, associative and distributive. - Statement 4 - Incorrect.
Convolution is applicable only to time-invariant systems and not to time-varying systems.
The correct principle is: Convolution is commutative, associative and distributive.
Core Concept
For an LTI system, output is the convolution of input and impulse response. Convolution is commutative, associative and distributive. Graphically, one signal is reversed, shifted, multiplied and summed/integrated. For finite discrete sequences of lengths Lx and Lh, full convolution length is Lx+Lh−1.
Formula / Key Relationship
y=x*h; DT: y[n]=Σ x[k]h[n−k].
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
Flip, shift, multiply, sum.
Common Mistake
Convolution usually increases finite-sequence length; it does not always reduce it.
Quick Revision
Properties of Convolution is linked with Convolution, LTI Systems and Signals and Systems. Flip, shift, multiply, sum.
With respect to the mathematical properties of convolution in linear time-invariant (LTI) systems, consider the following statements:
1. The convolution operation between two signals satisfies the commutative property irrespective of signal type.
2. The associative property of convolution allows grouping of signals without affecting the final output.
3. Convolution distributes over addition, enabling decomposition of complex inputs into simpler components.
4. Convolution is applicable only to time-invariant systems and not to time-varying systems.
Which of the above statements is/are correct?
Answer & step-by-step solution
For an LTI system, output is the convolution of input and impulse response. Convolution is commutative, associative and distributive. Key relation: y=x*h; DT: y[n]=Σ x[k]h[n−k].
Exam focus: Flip, shift, multiply, sum. Common trap: Convolution usually increases finite-sequence length; it does not always reduce it.
Statement-wise Verification
- Statement 1 - Correct.
The convolution operation between two signals satisfies the commutative property irrespective of signal type.
Convolution is commutative, associative and distributive. - Statement 2 - Correct.
The associative property of convolution allows grouping of signals without affecting the final output.
Convolution is commutative, associative and distributive. - Statement 3 - Correct.
Convolution distributes over addition, enabling decomposition of complex inputs into simpler components.
Convolution is commutative, associative and distributive. - Statement 4 - Incorrect.
Convolution is applicable only to time-invariant systems and not to time-varying systems.
The correct principle is: Convolution is commutative, associative and distributive.
Core Concept
For an LTI system, output is the convolution of input and impulse response. Convolution is commutative, associative and distributive. Graphically, one signal is reversed, shifted, multiplied and summed/integrated. For finite discrete sequences of lengths Lx and Lh, full convolution length is Lx+Lh−1.
Formula / Key Relationship
y=x*h; DT: y[n]=Σ x[k]h[n−k].
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
Flip, shift, multiply, sum.
Common Mistake
Convolution usually increases finite-sequence length; it does not always reduce it.
Quick Revision
Properties of Convolution is linked with Convolution, LTI Systems and Signals and Systems. Flip, shift, multiply, sum.
रैखिक समय-अपरिवर्ती (LTI) प्रणालियों में कनवल्यूशन के गणितीय गुणों के संबंध में निम्नलिखित कथनों पर विचार कीजिए:
1. दो संकेतों के बीच कनवल्यूशन संक्रिया संकेत प्रकार की परवाह किए बिना क्रमविनिमेय (commutative) गुण संतुष्ट करती है।
2. कनवल्यूशन का साहचर्य (associative) गुण अंतिम आउटपुट प्रभावित किए बिना संकेतों को समूहित करने देता है।
3. कनवल्यूशन योग पर वितरित होता है, जिससे जटिल इनपुट को सरल घटकों में विभाजित किया जा सकता है।
4. कनवल्यूशन केवल समय-अपरिवर्ती प्रणालियों पर लागू होता है, समय-परिवर्ती प्रणालियों पर नहीं।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
B — केवल 1, 2 और 3। कनवल्यूशन क्रमविनिमेय (x*h = h*x), साहचर्य ((x*h₁)*h₂ = x*(h₁*h₂)) और वितरणशील (x*(h₁+h₂) = x*h₁ + x*h₂) है। कनवल्यूशन एक गणितीय संक्रिया है जो समय-परिवर्ती संदर्भों में भी परिभाषित की जा सकती है, इसलिए कथन 4 गलत है।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
The convolution operation between two signals satisfies the commutative property irrespective of signal type.
Convolution is commutative, associative and distributive. - Statement 2 - Correct.
The associative property of convolution allows grouping of signals without affecting the final output.
Convolution is commutative, associative and distributive. - Statement 3 - Correct.
Convolution distributes over addition, enabling decomposition of complex inputs into simpler components.
Convolution is commutative, associative and distributive. - Statement 4 - Incorrect.
Convolution is applicable only to time-invariant systems and not to time-varying systems.
The correct principle is: Convolution is commutative, associative and distributive.
Core Concept
For an LTI system, output is the convolution of input and impulse response. Convolution is commutative, associative and distributive. Graphically, one signal is reversed, shifted, multiplied and summed/integrated. For finite discrete sequences of lengths Lx and Lh, full convolution length is Lx+Lh−1.
Formula / Key Relationship
y=x*h; DT: y[n]=Σ x[k]h[n−k].
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
Flip, shift, multiply, sum.
Common Mistake
Convolution usually increases finite-sequence length; it does not always reduce it.
Quick Revision
Properties of Convolution is linked with Convolution, LTI Systems and Signals and Systems. Flip, shift, multiply, sum.
More LTI Systems & Convolution questions